Finite Group Databasebeta

← #13321639

Subgroups — #13321639

Order 1792.

1010 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11742856C211yesmaximal, characteristic
2C7×#2353C211yesmaximal, characteristic, complemented
3#11739747C211yesmaximal, characteristic, complemented
4#11727320C211yesmaximal, characteristic, complemented
5#11727318C211yesmaximal, characteristic, complemented
6#11727286C212yesmaximal, complemented
7#11727262C212yesmaximal, complemented
8#11727197C212yesmaximal, complemented
9#11727187C211yesmaximal, characteristic, complemented
10#11727186C211yesmaximal, characteristic, complemented
11#11727152C212yesmaximal, complemented
12C2×C4×C7×D4C2211yescharacteristic
13#90496—21no—
14#90482C2212yes—
15#90482C2212yes—
16#90254C412yescomplemented
17#90250—22no—
18#90249—22no—
19#90244C412yescomplemented
20#90242C412yescomplemented
21#90233C412yescomplemented
22#90195—21no—
23#90185—21no—
24#90185—21no—
25#90168C2211yescharacteristic
26#90161C2211yescharacteristic
27C7×#388—21no—
28C7×#387—21no—
29C7×#386—21no—
30C7×#383C2211yescharacteristic
31C7×#381C2211yescharacteristic
32C7×#380C2211yescharacteristic
33C4×C7×C22⋊C4—21no—
34C4×C7×C22⋊C4C2211yescharacteristic
35C7×#376C2211yescharacteristic
36C2×C7×#95C2211yescharacteristic
37#89979C2212yes—
38C2×C22⋊(C7⋊C8)C2212yes—
39#89783—22no—
40#89783—22no—
41#89698C2211yescharacteristic
42#89698C2211yescharacteristic
43#89247C2212yes—
44#89247C2212yes—
45#89247C2212yes—
46#89247C2212yes—
47#89246C2212yes—
48#89246C2212yes—
49#89246C2212yes—
50#89246C2212yes—
51#17629—71nomaximal, Sylow
52C22×C7×D4D411yescharacteristic
53C23×C4×C7—21no—
54C23×C4×C7C2311yescharacteristic
55C23×C4×C7C2311yescharacteristic
56C2×#1157—21no—
57C2×#1157C2×C412yes—
58C2×#1157C2×C412yescomplemented
59C4×C7×D4C2×C412yes—
60C4×C7×D4C2×C412yes—
61C4×C7×D4D412yes—
62C4×C7×D4D412yes—
63C2×C7×C4⋊C4—21no—
64C2×C7×C4⋊C4D411yescharacteristic
65C2×C7×C22⋊C4—21no—
66C2×C7×C22⋊C4—21no—
67C2×C7×C22⋊C4—21no—
68C2×C7×C22⋊C4D411yescharacteristic
69C2×C7×C22⋊C4D411yescharacteristic
70C2×C42×C7—21no—
71C2×C42×C7—21no—
72C2×C42×C7C2311yescharacteristic
73#6516—22no—
74#6516—22no—
75#6503—22no—
76#6501—22no—
77#6501—22no—
78#6499—22no—
79#6493—21no—
80#6493C2×C412yes—
81#6493C2×C412yescomplemented
82#6493C2312yes—
83#6492—22no—
84#6492—22no—
85C4×C22⋊D7—22no—
86C4×C22⋊D7—22no—
87C4×C22⋊D7—22no—
88C4×C22⋊D7—22no—
89#6489—21no—
90#6489C2×C412yes—
91#6489C2×C412yescomplemented
92#6487—21no—
93#6487C2×C412yes—
94#6487C2×C412yescomplemented
95#6487C2312yes—
96C2×#1136—22no—
97C22×C7⋊C8—22no—
98#6456—22no—
99#6455—22no—
100D7×C4⋊C4—22no—
101#6450—22no—
102#6448—22no—
103#6448—22no—
104#6447—22no—
105#6447—22no—
106#6446—22no—
107#6446—22no—
108#6442—22no—
109#6442—22no—
110#6438—21no—
111#6437C2×C411yescharacteristic, complemented
112C4×D28—21no—
113C4×D28C2×C412yescomplemented
114#6433—21no—
115#6432C2×C411yescharacteristic, complemented
116#6432C2×C411yescharacteristic, complemented
117#6432C2×C411yescharacteristic, complemented
118C4×Dic14—21no—
119C4×Dic14C2×C412yescomplemented
120C7×#95—21no—
121C7×#95—21no—
122C7×#95—21no—
123C7×#95—21no—
124C7×#95—21no—
125C7×#95C2311yescharacteristic
126C7×#95C2311yescharacteristic
127C7×#95C2311yescharacteristic
128C7×#95C2311yescharacteristic
129C22⋊(C7⋊C8)—22no—
130C22⋊(C7⋊C8)—22no—
131C22⋊(C7⋊C8)C2312yes—
132C22⋊(C7⋊C8)C2312yes—
133C4⋊(C7⋊C8)—21no—
134C4⋊(C7⋊C8)—21no—
135#6377—21no—
136#6377—21no—
137#3506—71no—
138#2973—71no—
139#2353D711yescharacteristic, complemented
140#1516—72no—
141#1515—71no—
142#1514—71no—
143#1513—72no—
144#1506—71no—
145#1505—71no—
146#1504—72no—
147#1501—72no—
148C24×C7C2×D411yescharacteristic
149C24×C7C2×D411yescharacteristic
150C2×C7×D4—41no—
151C2×C7×D4—41no—
152C2×C7×D4C22⋊C412yes—
153C2×C7×D4C22⋊C412yes—
154C22×C4×C7—21no—
155C22×C4×C7—21no—
156C22×C4×C7—21no—
157C22×C4×C7—21no—
158C22×C4×C7—42no—
159C22×C4×C7—42no—
160C22×C4×C7—81no—
161C22×C4×C7—81no—
162C22×C4×C7—81no—
163C22×C4×C7C22×C412yes—
164C22×C4×C7C22×C412yes—
165C22×C4×C7C2×D411yescharacteristic
166C22×C4×C7C2×D411yescharacteristic
167C22×C4×C7C2×D411yescharacteristic
168C22×C4×C7C2×D411yescharacteristic
169C22×C4×C7C2×D412yes—
170C22×C4×C7C2×D412yes—
171C22×C4×C7C2411yescharacteristic
172C2×C22⋊D7—22no—
173C2×C22⋊D7—22no—
174C2×C22⋊D7—22no—
175C2×C22⋊D7—22no—
176C2×C22⋊D7—22no—
177#1157—42no—
178#1157—42no—
179#1157—82no—
180#1157—82no—
181C2×D28—21no—
182C2×D28—21no—
183C2×D28C22×C412yes—
184C2×C4×D7—22no—
185C2×C4×D7—22no—
186C2×C4×D7—22no—
187C2×Dic14—21no—
188C2×Dic14—21no—
189C2×Dic14C22×C412yes—
190C7×C4⋊C4—81no—
191C7×C4⋊C4C22⋊C412yes—
192C7×C4⋊C4C22⋊C412yes—
193C7×C22⋊C4—42no—
194C7×C22⋊C4—42no—
195C7×C22⋊C4—81no—
196C7×C22⋊C4C22⋊C412yes—
197C7×C22⋊C4C22⋊C412yes—
198C7×C22⋊C4C22⋊C412yes—
199C7×C22⋊C4C22⋊C412yes—
200C42×C7—42no—
201C42×C7—81no—
202C42×C7C22×C411yescharacteristic
203C42×C7C22×C411yescharacteristic
204C42×C7C2×D411yescharacteristic
205C42×C7C2×D411yescharacteristic
206C22⋊Dic7—22no—
207C22⋊Dic7—22no—
208C22⋊Dic7—22no—
209C22⋊Dic7—22no—
210C22⋊Dic7—22no—
211(C2×D7)⋊C4—22no—
212(C2×D7)⋊C4—22no—
213(C2×D7)⋊C4—22no—
214(C2×D7)⋊C4—22no—
215C4⋊Dic7—21no—
216C4⋊Dic7—21no—
217C4⋊Dic7—21no—
218C4⋊Dic7—21no—
219C4⋊Dic7C22×C412yes—
220C4⋊Dic7C22×C412yes—
221Dic7⋊C4—22no—
222Dic7⋊C4—22no—
223Dic7⋊C4—22no—
224Dic7⋊C4—22no—
225C4×Dic7—22no—
226C4×Dic7—22no—
227C4×Dic7—22no—
228#1136—82no—
229C2×C7⋊C8—22no—
230C2×C7⋊C8—22no—
231C2×C7⋊C8—82no—
232#559—142no—
233#556—72no—
234#555—72no—
235#553—142no—
236#550—72no—
237#547—72no—
238#538—141no—
239#538—141no—
240#535—141no—
241#533—71no—
242#532—71no—
243#525—72no—
244#525—72no—
245#520—141no—
246C2×C4×D4C2×D711yescharacteristic
247#435—142no—
248#435—142no—
249#432—71no—
250#432—71no—
251#407—72no—
252C2×C22⋊C8—72no—
253#388—21no—
254#387—21no—
255#386—21no—
256#383C2×D711yescharacteristic
257#381C2×D711yescharacteristic
258#380C2×D711yescharacteristic
259C4×C22⋊C4—21no—
260C4×C22⋊C4C2×D711yescharacteristic
261#376C2×D711yescharacteristic
262C2×#95C2×D711yescharacteristic
263#328—72no—
264#328—72no—
265#328—72no—
266#328—72no—
267#327—72no—
268#327—72no—
269#327—72no—
270#327—72no—
271C23×C7—21no—
272C23×C7—21no—
273C23×C7—21no—
274C23×C7—21no—
275C23×C7—41no—
276C23×C7—41no—
277C23×C7—41no—
278C23×C7—41no—
279C23×C7C2×C22⋊C412yes—
280C23×C7C2×C22⋊C412yes—
281C23×C7C22×D411yescharacteristic
282C22×D7—22no—
283C7×D4—42no—
284C7×D4—42no—
285C2×C4×C7—21no—
286C2×C4×C7—21no—
287C2×C4×C7—22no—
288C2×C4×C7—42no—
289C2×C4×C7—42no—
290C2×C4×C7—41no—
291C2×C4×C7—41no—
292C2×C4×C7—41no—
293C2×C4×C7—42no—
294C2×C4×C7—42no—
295C2×C4×C7—41no—
296C2×C4×C7—41no—
297C2×C4×C7—41no—
298C2×C4×C7—41no—
299C2×C4×C7—41no—
300C2×C4×C7—41no—
301C2×C4×C7—41no—
302C2×C4×C7—41no—
303C2×C4×C7—41no—
304C2×C4×C7—41no—
305C2×C4×C7—41no—
306C2×C4×C7—41no—
307C2×C4×C7—84no—
308C2×C4×C7C2×C22⋊C411yescharacteristic
309C2×C4×C7C2×C22⋊C412yes—
310C2×C4×C7C2×C22⋊C412yes—
311C2×C4×C7C2×C22⋊C411yescharacteristic
312C2×C4×C7C2×C22⋊C411yescharacteristic
313C2×C4×C7C2×C22⋊C411yescharacteristic
314C2×C4×C7C23×C411yescharacteristic
315C2×C4×C7C22×D411yescharacteristic
316C22⋊D7—42no—
317C22⋊D7—42no—
318C22⋊D7—82no—
319C22⋊D7—82no—
320C22⋊D7—82no—
321C22⋊D7—82no—
322C2×Dic7—22no—
323C2×Dic7—22no—
324C2×Dic7—22no—
325D28—42no—
326D28—42no—
327D28—81no—
328C4×D7—44no—
329C4×D7—82no—
330C4×D7—82no—
331Dic14—42no—
332Dic14—42no—
333Dic14—81no—
334C7⋊C8—42no—
335C7⋊C8—42no—
336C2×#41—72no—
337C2×#41—72no—
338C2×#41—141no—
339C22×D4C22⋊D711yescharacteristic
340C23×C4—21no—
341C23×C4C22×D711yescharacteristic
342C23×C4C22×D711yescharacteristic
343C2×M16—142no—
344C22×C8—142no—
345C4⋊Q8—71no—
346C4⋊Q8—71no—
347C4⋊Q8—71no—
348#127—71no—
349#127—142no—
350#126—142no—
351#126—142no—
352#125—141no—
353#124—142no—
354#124—142no—
355#124—142no—
356#124—141no—
357#123—142no—
358#123—142no—
359#123—142no—
360#123—142no—
361C22⋊Q8—72no—
362C22⋊Q8—72no—
363C22⋊Q8—72no—
364C22⋊Q8—141no—
365#121—72no—
366#121—72no—
367#121—72no—
368#121—142no—
369#121—142no—
370#121—141no—
371#121—142no—
372#121—142no—
373#120—142no—
374#120—142no—
375C4×Q8—72no—
376C4×Q8—142no—
377C4×Q8—141no—
378C4×D4—72no—
379C4×D4—142no—
380C4×D4—142no—
381C4×D4—142no—
382C4×D4—142no—
383C4×D4—142no—
384C4×D4—141no—
385C4×D4C4×D712yes—
386C4×D4C4×D712yes—
387C4×D4D2812yes—
388C4×D4D2812yes—
389#117—72no—
390#117—72no—
391#117—141no—
392#117—142no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—142no—
395C2×C4⋊C4C22⋊D711yescharacteristic
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C22⋊C4C22⋊D711yescharacteristic
400C2×C22⋊C4C22⋊D711yescharacteristic
401C2×C42—21no—
402C2×C42—21no—
403C2×C42C22×D711yescharacteristic
404C4⋊C8—141no—
405C4⋊C8—141no—
406C22⋊C8—72no—
407C22⋊C8—72no—
408C22⋊C8—142no—
409C22⋊C8—142no—
410#97—141no—
411#97—141no—
412#95—21no—
413#95—21no—
414#95—21no—
415#95—21no—
416#95—21no—
417#95C22×D711yescharacteristic
418#95C22×D711yescharacteristic
419#95C22×D711yescharacteristic
420#95C22×D711yescharacteristic
421C22×C7—21no—
422C22×C7—21no—
423C22×C7—22no—
424C22×C7—22no—
425C22×C7—22no—
426C22×C7—22no—
427C22×C7—22no—
428C22×C7—22no—
429C22×C7—42no—
430C22×C7—42no—
431C22×C7—81no—
432C22×C7—81no—
433C22×C7C22×C22⋊C411yescharacteristic
434C22×C7#57511yescharacteristic
435C22×C7#57511yescharacteristic
436C2×D7—42no—
437C2×D7—42no—
438C2×D7—42no—
439C4×C7—21no—
440C4×C7—21no—
441C4×C7—22no—
442C4×C7—22no—
443C4×C7—21no—
444C4×C7—21no—
445C4×C7—21no—
446C4×C7—21no—
447C4×C7—42no—
448C4×C7—81no—
449C4×C7—81no—
450C4×C7—81no—
451Dic7—42no—
452Dic7—42no—
453Dic7—42no—
454C24C2×C22⋊D711yescharacteristic
455C24C2×C22⋊D711yescharacteristic
456#41—282no—
457#41—282no—
458#41—562no—
459#41—562no—
460C2×Q8—72no—
461C2×Q8—141no—
462C2×Q8—141no—
463C2×D4—41no—
464C2×D4—41no—
465C2×D4—72no—
466C2×D4—142no—
467C2×D4—142no—
468C2×D4—142no—
469C2×D4—142no—
470C2×D4—142no—
471C2×D4—141no—
472C2×D4—141no—
473C2×D4(C2×D7)⋊C412yes—
474C2×D4(C2×D7)⋊C412yes—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—81no—
482C22×C4—81no—
483C22×C4—81no—
484C22×C4—142no—
485C22×C4—142no—
486C22×C4—142no—
487C22×C4C2×C4×D712yes—
488C22×C4C2×C4×D712yes—
489C22×C4C2×D2812yes—
490C22×C4C2×D2812yes—
491C22×C4C2×C22⋊D711yescharacteristic
492C22×C4C2×C22⋊D711yescharacteristic
493C22×C4C2×C22⋊D711yescharacteristic
494C22×C4C2×C22⋊D711yescharacteristic
495C22×C4C23×D711yescharacteristic
496M16—562no—
497C2×C8—142no—
498C2×C8—142no—
499C2×C8—562no—
500C4⋊C4—72no—
501C4⋊C4—72no—
502C4⋊C4—81no—
503C4⋊C4—142no—
504C4⋊C4—142no—
505C4⋊C4—142no—
506C4⋊C4—141no—
507C4⋊C4—141no—
508C4⋊C4—141no—
509C4⋊C4—141no—
510C4⋊C4—142no—
511C4⋊C4(C2×D7)⋊C412yes—
512C4⋊C4(C2×D7)⋊C412yes—
513C22⋊C4—42no—
514C22⋊C4—42no—
515C22⋊C4—81no—
516C22⋊C4—142no—
517C22⋊C4—142no—
518C22⋊C4—142no—
519C22⋊C4—142no—
520C22⋊C4—142no—
521C22⋊C4—142no—
522C22⋊C4—142no—
523C22⋊C4—142no—
524C22⋊C4—142no—
525C22⋊C4(C2×D7)⋊C412yes—
526C22⋊C4(C2×D7)⋊C412yes—
527C22⋊C4(C2×D7)⋊C412yes—
528C22⋊C4(C2×D7)⋊C412yes—
529C42—42no—
530C42—81no—
531C42—142no—
532C42—142no—
533C42—142no—
534C42C2×C4×D711yescharacteristic
535C42C2×C4×D711yescharacteristic
536C42C2×D2811yescharacteristic
537C42C2×D2811yescharacteristic
538C2×C7—21no—
539C2×C7—21no—
540C2×C7—42no—
541C2×C7—42no—
542C2×C7#288311yescharacteristic
543C2×C7#289611yescharacteristic
544C2×C7#290111yescharacteristic
545D7—82no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—142no—
555C23C2×(C2×D7)⋊C412yes—
556C23C2×(C2×D7)⋊C412yes—
557C23C22×C22⋊D711yescharacteristic
558Q8—282no—
559Q8—282no—
560Q8—561no—
561D4—42no—
562D4—42no—
563D4—282no—
564D4—282no—
565D4—282no—
566D4—282no—
567D4—561no—
568D4—562no—
569D4—562no—
570D4—562no—
571D4—562no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—84no—
595C2×C4—142no—
596C2×C4—142no—
597C2×C4—142no—
598C2×C4—284no—
599C2×C4—562no—
600C2×C4—562no—
601C2×C4C2×(C2×D7)⋊C411yescharacteristic
602C2×C4C2×(C2×D7)⋊C412yes—
603C2×C4C2×(C2×D7)⋊C412yes—
604C2×C4C2×(C2×D7)⋊C411yescharacteristic
605C2×C4C2×(C2×D7)⋊C411yescharacteristic
606C2×C4C2×(C2×D7)⋊C411yescharacteristic
607C2×C4C22×C4×D711yescharacteristic
608C2×C4C22×D2811yescharacteristic
609C8—282no—
610C8—282no—
611C7#1762911yescharacteristic, complemented, Sylow
612C22—21no—
613C22—21no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—42no—
621C22—42no—
622C22—81no—
623C22—81no—
624C22—282no—
625C22—282no—
626C22—282no—
627C22C22×(C2×D7)⋊C411yescharacteristic
628C22#9051311yescharacteristic
629C22#9051311yescharacteristic
630C4—21no—
631C4—21no—
632C4—21no—
633C4—21no—
634C4—21no—
635C4—21no—
636C4—22no—
637C4—22no—
638C4—42no—
639C4—81no—
640C4—81no—
641C4—81no—
642C4—282no—
643C4—282no—
644C4—282no—
645C2—21no—
646C2—21no—
647C2—42no—
648C2—42no—
649C2—562no—
650C2#1173965811yescharacteristic
651C2#1173998111yescharacteristic
652C2#1173999111yescharacteristic
subgroupG/H
centerC22C22×(C2×D7)⋊C4
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#2353C2
socleC22×C7C22×C22⋊C4
prime radical#11742856C2
cosocleC22×C4×C7C24
hypercenter#2353D7
hypocenterC7#17629
LCSG ▷ C2×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2353

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C7×#2353C2semidirect
3. #11739747C2semidirect
4. #11727320C2semidirect
5. #11727318C2semidirect
6. #11727286C2semidirect
7. #11727262C2semidirect
8. #11727197C2semidirect
9. #11727187C2semidirect
10. #11727186C2semidirect
11. #11727152C2semidirect
16. #90254C4semidirect
19. #90244C4semidirect
20. #90242C4semidirect
21. #90233C4semidirect
58. C2×#1157C2×C4semidirect
81. #6493C2×C4semidirect
91. #6489C2×C4semidirect
94. #6487C2×C4semidirect
111. #6437C2×C4semidirect
113. C4×D28C2×C4semidirect
115. #6432C2×C4semidirect
116. #6432C2×C4semidirect
117. #6432C2×C4semidirect
119. C4×Dic14C2×C4semidirect
139. #2353D7semidirect
611. C7#17629semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#117428562
C7×#23532
#117397472
#117273202
#117273182
#117272862
#117272622
#117271972
#117271872
#117271862
#117271522
#176297